This paper considers an energy-limited cognitive relay network where a secondary transmitter (ST) assists to forward the traffic from a primary transmitter (PT) to a primary receiver (PR), in exchange for serving its own secondary receiver (SR) in the same frequency. The multiple-antenna ST is assumed to be energy-constrained and powered by both information flow from source (PT) and dedicated energy streams from destinations (PR and SR), which is called destination-aided wireless power transfer (DWPT) scheme. Then, the relay processing matrix, cognitive beamforming vector and power splitter are jointly de- signed to maximize the rate of secondary users under the energy causality constraint and the constraint that the demanded rate of primary users is satisfied. For the perfect channel information state (CSI) case, by adopting semi-definite relax (SDR) technique and Charnes-Cooper transformation, the global optimal solution is given. To reduce the complexity, matrix decomposition, zero forcing (ZF) scheme, and dual method are jointly employed to derive a suboptimal solution. For the imperfect CSI case, S- procedure is used to transform the worst-case robust problem into a tractable semi-definite program (SDP). Simulation results reveal that our proposed DWPT scheme is greatly preferred for both perfect and imperfect CSI cases when ST is close to PR/SR.